[Paper Review] Orbital stability of standing waves of a class of fractional Schrodinger equations with a general Hartree-type integrand

Y. Cho, Mouhamed Moustapha Fall+3 more|arXiv (Cornell University)|Jul 21, 2013

Advanced Mathematical Physics Problems15 references5 citations

TL;DR

This paper establishes the existence, uniqueness, and orbital stability of standing wave solutions for a class of nonlinear fractional Schrödinger equations with a general Hartree-type nonlinearity. Using concentration-compactness and variational methods, it proves global well-posedness and shows that minimizers of the associated energy constraint problem are orbitally stable under suitable conditions on the nonlinearity and potential parameters.

ABSTRACT

This article is concerned with the mathematical analysis of a class of a nonlinear fractional Schrodinger equations with a general Hartree-type integrand. We prove existence and uniqueness of global-in-time solutions to the associated Cauchy problem. Under suitable assumptions, we also prove the existence of standing waves using the method of concentration-compactness by studying the associated constrained minimization problem. Finally we show the orbital stability of standing waves which are the minimizers of the associate variational problem.

Motivation & Objective

• To establish the existence and uniqueness of global-in-time solutions to the Cauchy problem for a fractional Schrödinger equation with a general Hartree-type nonlinearity.
• To investigate the existence of standing wave solutions by solving a constrained minimization problem in the energy space $ H^s(\mathbb{R}^N) $.
• To prove the orbital stability of standing wave solutions that minimize the associated variational problem.
• To extend previous results on boson stars and massless Schrödinger equations to a broader class of nonlinearities and fractional powers.
• To analyze the regularity and compactness properties of minimizing sequences to ensure stability under perturbations.

Proposed method

• Employ the concentration-compactness principle to analyze minimizing sequences for the constrained variational problem associated with the energy functional.
• Use the fractional Laplacian $ (-\Delta)^s $ defined via the Fourier transform as $ \mathcal{F}[(-\Delta)^s \phi](\xi) = |\xi|^{2s} \mathcal{F}[\phi](\xi) $, with $ 0 < s < 1 $.
• Model the nonlinearity via $ G(|\phi|) \star V(|x|) $, where $ V(|x|) = |x|^{\beta - N} $ with $ \beta > 0 $, $ \beta > N - 2s $, and $ G: \mathbb{R}^+ \to \mathbb{R}^+ $ is differentiable with $ G(0) = 0 $.
• Apply the Hardy-Littlewood-Sobolev inequality and regularity theory for Riesz potentials to establish local Hölder and Hölder-differentiable regularity of solutions.
• Prove relative compactness of minimizing sequences in $ H^s(\mathbb{R}^N) $, which is essential for orbital stability.
• Use energy and mass conservation laws for the Cauchy problem to support the stability analysis of standing waves.

Experimental results

Research questions

• RQ1 Under what conditions does the Cauchy problem for the fractional Schrödinger equation with Hartree-type nonlinearity admit a global-in-time solution?
• RQ2 Do minimizers of the constrained energy functional exist, and are they radial and decreasing under general assumptions on $ G $ and $ \beta $?
• RQ3 Can the orbital stability of standing wave solutions be established when they arise as minimizers of the variational problem?
• RQ4 How does the regularity of the potential $ V(|x|) = |x|^{\beta - N} $ and the nonlinearity $ G $ affect the regularity and stability of solutions?
• RQ5 What are the necessary and sufficient conditions on the parameters $ s, \beta, N $, and the growth exponent $ \mu $ in $ \mathcal{A}_0 $ for uniqueness and stability?

Key findings

• The Cauchy problem admits a weak global-in-time solution $ \phi \in L^\infty(\mathbb{R}; H^s(\mathbb{R}^N)) \cap W^{1,\infty}(\mathbb{R}; H^{-s}(\mathbb{R}^N)) $ under the condition $ \mathcal{A}_0 $ on $ G $, with $ \phi_0 \in H^s(\mathbb{R}^N) $.
• Under specific conditions on $ s, \beta, N $, and $ \mu $, the solution is unique, particularly when $ \mu \in \left(\max\left(2, 1 + \frac{2\beta - N}{N - 2s}\right), 2 + \frac{N}{N - 2s} \cdot \frac{2s - 1 - 2N + 2\beta}{2s - 1 + N}\right) $.
• Minimizers of the variational problem $ \mathcal{I}^G_\lambda $ exist and are radial and decreasing under the given assumptions on $ G $, $ \beta $, and $ s $.
• The standing wave solutions, which are minimizers of the energy functional under $ L^2 $-norm constraint, are orbitally stable in the sense that small perturbations in $ H^s $ norm remain close to the orbit of the solution.
• The potential $ V \star G(\phi) $ is shown to be in $ C^{0,\alpha}_{\text{loc}}(\mathbb{R}^N) $ for $ \beta \leq 1 $ and $ C^{[\beta],\alpha}_{\text{loc}}(\mathbb{R}^N) $ for $ \beta > 1 $, ensuring sufficient regularity for the nonlinear term.
• The solution $ u $ to the stationary equation is in $ C^{0,\alpha}_{\text{loc}}(\mathbb{R}^N) $ for $ \beta \leq 1 $ and $ C^{1,\alpha}_{\text{loc}}(\mathbb{R}^N) $ for $ \beta > 1 $, following from regularity propagation through the nonlinearity.